1. Trim the chicken of excess fat. Place in a large bowl and sprinkle with salt and the vinegar. Set aside.

2. Heat the olive oil in a large pot over medium heat and add the onions. Reduce the heat to very low, cover the pot, and let the onions steam in the oil for about 25 minutes. Check and stir occasionally to keep them from burning or sticking to the bottom of the pot.

3. Drain the chicken and pat dry with paper towels. Remove the onions from the pot with a slotted spoon, leaving as much olive oil behind as possible. Raise the heat a little and brown the chicken pieces, in batches if necessary. Remove from the pot and set aside. Return the onions to the pot, place the chicken pieces on top, and season with salt and pepper. Stir a little so that the onions are all around and on top of the chicken. Cover, reduce the heat to very low, and simmer until the chicken is very well cooked and almost falling off the bone, about 1 hour. During cooking, add a little water if necessary to keep the chicken and onions from burning. Add the crumbled feta to the pot. Cover and cook until the feta is completely melted and the sauce is very thick, another 8 minutes or so. Remove from the heat and serve.